I Separation of Variables (PDE) for the Laplace Equation

  • Thread starter FAS1998
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Summary
I've attached an image of a solved problem. Can somebody explain the steps in the yellow box? I don't understand how they got to that point from the previous steps.
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phyzguy

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I suspect it is the first step that is bothering you.
[tex] \sum_{n=1}^{\infty}a_n \sin(\frac{n \pi x}{2}) \sinh(\frac{n \pi}{2}) = \sin(\frac{\pi x}{2}) [/tex]
This is true for all values of x. The only way this can be true is for all of the [itex] a_n[/itex] to be zero except [itex]a_1[/itex]. Is this the step that is troubling you? After this, the rest follows pretty easily.
 
40
1
I suspect it is the first step that is bothering you.
[tex] \sum_{n=1}^{\infty}a_n \sin(\frac{n \pi x}{2}) \sinh(\frac{n \pi}{2}) = \sin(\frac{\pi x}{2}) [/tex]
This is true for all values of x. The only way this can be true is for all of the [itex] a_n[/itex] to be zero except [itex]a_1[/itex]. Is this the step that is troubling you? After this, the rest follows pretty easily.
That is the step that is bothering me. Can you explain why all values of a must be 0 except a1?
 

phyzguy

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1,222
That is the step that is bothering me. Can you explain why all values of a must be 0 except a1?
The functions [itex] \sin(\frac{n \pi x}{2})[/itex] are orthogonal functions on the interval (0,2). If I integrate:
[tex] \int_0^2 \sin(\frac{n \pi x}{2}) \sin(\frac{m \pi x}{2}) dx = \delta_{nm} [/tex]
so it is zero unless n=m. So if you take your original expression, multiply both sides by [itex] \sin(\frac{m \pi x}{2})[/itex] and integrate both sides you get:

[tex] \sum_{n=1}^{\infty}a_n \sinh(\frac{n \pi}{2}) \int_0^2 \sin(\frac{n \pi x}{2}) \sin(\frac{m \pi x}{2}) = \int_0^2 \sin(\frac{\pi x}{2}) \sin(\frac{m \pi x}{2}) dx [/tex]

This gives [itex] a_m \sinh(\frac{m \pi}{2}) = 0 [/itex] if m is not equal to 1, and [itex] a_m \sinh(\frac{m \pi}{2}) = 1 [/itex] if m = 1.
 

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